Magnetically tunable graphene-based reflector under linear polarized incidence at room temperature

Graphene magnetoplasmons (GMPs) are the collective oscillations of two-dimensional massless electrons in graphene under magnetostatic bias.^1–3 At the terahertz (THz) spectrum, graphene exhibits nonreciprocal and gyrotropic responses due to GMPs.⁴ This can be characterized by the anisotropic conductivity with a two-dimensional tensor derived from the Kubo formula.⁵ The electrostatic tunable conductivity and carrier-density-dependent cyclotron mass of graphene make it different from conventional gyromagnetic materials. In recent years, applications in health, security, astronomy, and communication operating in the THz spectrum have begun to emerge.⁶ Graphene has shown potential to be employed in novel magneto-optical devices for THz applications.⁷ To provide a roadmap for various graphene-based modulators and non-reciprocal devices, researchers have proposed fundamental limits to estimate their optimal performance.⁸ 

In terms of nonreciprocal performance at the THz spectrum, graphene-based isolators, demonstrating up to 20 dB isolation, have been experimentally demonstrated for circularly polarized wave incidence.⁹ Tunable isolation performance can be achieved by changing the carrier density of graphene.¹⁰ Moreover, the gyrotropic properties of GMPs have also been demonstrated for Faraday rotations (FRs) at room temperature.¹¹ The FR angle can be enhanced through the effects of Fabry-Perot resonances.¹² However, the tradeoff between the FR angle and the transmission coefficient needs to be considered in such enhancement.¹³ Except for the FRs, Kerr rotations (KRs) in reflected waves from graphene have also been measured at low temperature (5 K).¹⁴ It is noticed that graphene conductivity is temperature-dependent with higher resistivity at higher temperature.^15,16 However, related applications of GMPs on the reflection configuration under linearly polarized illuminations at room temperature have not been reported in great detail.

In this letter, measurement results of a graphene-based reflector at room temperature under magnetostatic bias are presented. The absolute reflection intensity is measured with linearly polarized EM wave incidence at 2.5–6 THz. A maximum tunability of 15% is observed in the intensity modulation depth (IMD) when a magnetic field of 6 T is applied to the sample. A modified equivalent circuit model (ECM) is used to analyze the IMD of the graphene-based reflector. Moreover, due to the gyrotropic response of graphene under magnetostatic bias, the reflector is capable of manipulating the polarization of reflected waves, known as Kerr rotation. By comparing the IMD with the performance on Kerr rotations (KRs) obtained from full-wave simulations, it is found out that the IMD will be reduced from 69% to 40% in order to achieve the best performance on KRs.

The schematic view of a bi-layer graphene-based reflector presented in this work is shown in Fig. 1(a). Monolayer graphene was grown on Cu foil (99.8% purity) by chemical vapour deposition (CVD).¹⁷ Once the Cu foil was loaded into the tube furnace (Graphene Square), it was annealed in H₂ (flow: 20  sccm) at $940$ °C for 30 min. After annealing, a uniform layer of graphene was grown on the Cu surface using a mixture of CH₄ and H₂ (flow: 4.6 and 20 sccm) for 30 min while keeping the temperature at $940$ °C. Graphene was then transferred onto the glass/indium tin oxide (ITO) substrate with thickness h = 0.14 mm by wet transfer.¹⁸ The ammonium persulfate solution was used for the chemical etching of Cu and polymethyl methacrylate (PMMA) was used as a sacrificial layer to support graphene during the etching process. After transfer, PMMA was dissolved in acetone. The transfer process was repeated once for the transfer of the second graphene layer onto the glass/ITO substrate.

The conductivity of two-stacked monolayer CVD graphene (i.e., bilayer CVD graphene) can be 2–10 times larger than monolayer graphene when positioned on a glass substrate, as the top layer of the bilayer graphene does not make contact with the underlying substrate.¹⁹ Assuming that the interaction between each layer is weak, the conductivity tensor σ_b characterizing the bilayer graphene in the x − y plane can be expressed as

where N is a value within the range of 2 to 10 representing the increment of overall conductivity compared with monolayer graphene and σ_d (σ_o) is the diagonal (off-diagonal) element in the conductivity tensor σ_m of monolayer graphene. At room temperature, σ_d and σ_o can be expressed, respectively, as^20,21

According to the equivalent circuit model (ECM) proposed for wave transmission through graphene,²² a modified ECM, as shown in Figs. 1(b) and 1(c), was utilized to calculate the reflection performance of the graphene-based reflector. In the modified ECMs, the glass substrate is characterised by its characteristic admittance Y_d and propagation constant γ_d which are calculated according to published data in the literature.²³ The ground ITO film is modelled as an equivalent sheet admittance Y_ITO.²⁴ 

The coupling relation between the voltage V₁ and V₂ in Fig. 1(c) can be expressed as

where $Ybd=N×σd$ (⁠$Ybo=N×σo$⁠) is the diagonal (off-diagonal) admittance of the bilayer graphene, $Y0≈1/377$ S is the intrinsic admittance of free space, and Y_sub is the input admittance of substrate expressed as

According to Eqs. (2)–(4), the sample input admittance Y_sample in Fig. 1(b) can be written as

Finally, intensity modulation depth (IMD) is defined as

where $RECM(B0)=(Y0−Ysample)/(Y0+Ysample)$ is the reflection coefficient of the graphene-based modulator under different magnetostatic biases.

In measurements, the Fourier-transform infrared spectroscopy (FTIR) setup with magnetostatic bias is employed as shown in Fig. 2. Linear incidence was generated by a randomly polarized Globar light source with a grid-wire gold polarizer positioned in front of it. The incidence illuminated the sample through a Michelson interferometer configuration. The spectrum was swept from 2.5 THz to 6 THz. The absolute intensity of reflection I_R from the sample was measured by a He-cooled bolometer detector. The detector and source were tilted by 2.5° from the normal to allow for the positioning of both instruments. Considering the fact that the angles were quite small, the experimental results can be treated as the reflection of a normal incidence. A split-coil superconducting magnet at low temperature generated a static magnetic bias in the direction vertical to the sample which was at room temperature. The IMD can be calculated as $1−IR(B0≠0)/IR(B0=0)$⁠. However, the formula, $1−[IR(B0≠0)/Iref(B0≠0)]/[IR(B0=0)/Iref(B0=0)]$⁠, which divides by the reflection intensity of a gold mirror $Iref(B0)$ as a reference measured under the same experimental conditions, is taken to remove any variation in the measurement system due to the generation of the magnetic field.

The experimental results of IMD based on the formula mentioned above are shown in Fig. 3. Due to the loss of the substrate, the spectra in Fig. 3 do not clearly show resonance behaviour. However, the tunability of graphene and the effect on the IMD are still demonstrated by the measured data. The experimental results are fitted according to the ECM. The relevant ECM parameters are as follows: N = 8, $μc=0.21$ eV, and Γ = 50 meV, corresponding to the mobility of 3923 cm²(V s).²⁵ This is consistent with the mobility of graphene 4050 cm² (V s) fabricated with the same method.¹⁷ It can be found that the variation of IMD is larger at lower frequencies, consistent with the fact that Dirac fermions in continuous graphene have a stronger magneto response at lower THz frequencies. At around 3 THz, a 15% IMD was measured with $B0=6$ T. The IMD increases with the increase in magnetostatic bias. According to Eq. (2), the magneto-conductivity of graphene is affected by B₀ through the cyclotron frequency ω_c. Because B₀ and μ_c play inverse roles in the value of ω_c, the IMD can be increased by reducing μ_c. Figure 4 shows the IMD obtained with ECM for $μc=$ 0.1–0.4 eV and $B0=6$ T. It can be found that higher IMD can be achieved with lower chemical potential. The maximum IMD of 69% is achieved at $μc=$ 0.1 eV.

On the other hand, the Kerr rotation caused by the magnetostatic response of graphene is also analyzed for $μc=0.1–0.4$ eV. The figure-of-merit (FOM) of Kerr rotation can be defined as $θR$⁠, where θ is the Kerr rotation angle and R is the reflection coefficient,²⁶ which is simulated with commercial multi-physics solver, COMSOL, and the calculation results are plotted in Fig. 5 for $B0=6$ T. It is easy to see that the highest FOM is achieved with $μc=0.15$ eV at 2.875 THz. However, the corresponding IMD, as shown in Fig. 4, is only 40%. Hence, the maximum IMD and FOM cannot be obtained at the same time.

In this work, the reflection performance of a graphene-based reflector under magnetostatic bias at room temperature is presented. The measurement results show the maximum variation of 15% in IMD when tuning the magnetostatic bias. In order to analyze the reflection performance, a modified ECM was derived, and it was shown how the tunability of graphene under magnetostatic bias is highly dependent on the chemical potential. The tradeoff between IMD and Kerr rotation has been discussed with the data obtained from full-wave simulations. Although the reflector is demonstrated under $B0=6$ T in this letter, the required magnetostatic bias is expected to be reduced utilizing patterned graphene.²⁷ The work is useful for potential graphene-based applications at room temperature.

The authors would like to thank Dr. Antonio Lombardo at University of Cambridge for the help with sample preparation. The authors would also like to acknowledge financial support from China Scholarship Council (CSC), the Engineering and Physical Sciences Research Council (EPSRC) (EP/K01711X/1), the EU Graphene Flagship (FP7-ICT-604391), and Graphene Core 1 (H2020 696656).